Does the law require that parents with minor children or workers with other family responsibilities be provided with flexible time or part time work options?

Yes

More about paternity leave:

Russian legislation does not provide for special paternity leave. Nevertheless, the Labour Code establishes certain guarantees to fathers on the birth of a child: leave for childcare may be used in whole or in part also by the child's father. In addition, the employer is obliged on the basis of a written application to provide an employee with leave without pay in case of birth of a child - up to five days.

Sources: §128 and 256 of the Labour Code

More about parental leave:

The parental leave is available for parents including adoptive parents (as well as carers) until a child reaches the age of three years. The parental leave is provided at the request of the employee. During this leave, parents are eligible for allowances and compensation. The parental leave is a family entitlement, thus either parent may take all the leave or they may divide the leave in parts.

The first child allowance is paid (by the Social Insurance Fund) until the child attains the age of one and a half years (18 months). It is 40% of the average earnings of the insured person. In the case of caring for two or more children until they attain the age of one and a half years (18 months), the monthly insurance allowances are summarized. This summarized amount of the allowances must not exceed 100% of the average earnings. Minimum rates are established by the federal law ‘About state benefits for citizens with children’ (in mind adjustment for inflation).

For those, who have not right to get insurance child allowance, there is state child benefit. It is also paid until the child attains the age of one and a half years (18 months). The federal law ‘About state benefits for citizens with children’ establishes rates of this benefit: for one group it is fixed (since February 2016 - 2908.62 rubles for first child and 5817.24 rubles for next children), for other it is connected with a percentage of previous average earnings (40% of average pay). There is also a provision for monthly maximum: since February 2016 it is 11634.50 rubles.

As for leave in respect of children (aged 18 months to 36 months), no benefit is payable through Social Insurance Fund rather the employer is required to pay a monthly sum of 50 roubles. Also it is necessary to take into account the regional coefficients for calculating the rate of the child allowance. This allowance is available for parents, other relatives, guardians, actually caring for a child.

Other types of child allowance are established by regional legislation.

Employment of a worker is secure during parental leave and workers have the right to return to their employment after taking parental leave. Employees having a child with disability are entitled to four days of paid leave every month to care for the child. This leave may be taken by one parent or shared between them. Collective agreements may provide for unpaid leave of up to 14 calendar days per year for employees with two or more children under the age of 14 years; employees with a disabled child under the age of 18 years; and single parent(s) raising a child under 14 years. This leave is granted at the request of employee and can be combined with annual leave however it cannot be carried over to next year.

Sources: §256 of the Labour Code; §13-15 of the Federal law ‘About state benefits for citizens with children’ 19.05.1995 № 81-ФЗ; §11 of the Resolutions of the Government of the Russian Federation ‘On approval of the procedure for appointment and payment of monthly compensation to certain categories of citizens’ 03.11.1994 № 1206, § 11.1-11.2 of the Federal law ‘About compulsory social insurance against temporary disability and maternity’ 29.12.2006 № 255-ФЗ